Little-known fact: The late comedian Jackie Gleason was good friends with President Richard Nixon. Even weirder? Both men were obsessed with UFOs.

This new book by Larry Holcombe — who’s been studying UFOs for 50 years — does make some wild and thinly substantiated claims. But it’s too fun to pass up.

To wit: Gleason’s ex-wife told Esquire magazine that one night, after a day spent golfing, Nixon “showed up at Gleason’s home and whisked him away to Homestead Air Force Base, where they entered a hanger in a high-security area.” There, the two men supposedly gazed upon the wreckage of a UFO and its dead alien cargo.

Holcombe, a clear believer, breaks down every rumor about UFO visitation and the government’s “involvement” in it. There are many stories of sightings by high-level people, and it’s no secret that the government has looked into many of these reports over the years.

In October 1969, outside of a Lions Club meeting in Leary, Ga., Holcombe writes that Jimmy Carter and some fellow club members “noticed a strange light in the western sky.”

(Kennedy) had a great concern that UFOs could be mistaken by the US or the Soviet Union as a nuclear attack, and it is believed he discussed this issue with [Soviet leader Nikita] Khrushchev.

- Larry Holcombe

Carter later recalled one of the men saying, “ ‘Look over to the west!’ ” Carter allegedly said there was a bright light in the sky, which came closer, stopped, then suddenly “changed color to blue, then to red, then back to white.” The former president called to the aircraft a UFO — “but doesn’t believe it was extraterrestrial.”

While campaigning for president, Carter promised, “If elected . . . he would disclose what the government knew about the UFO phenomenon,” however, to Holcombe’s knowledge, he never did.

Ronald Reagan is regarded by UFO-logists as “the most open proponent of the existence of UFOs and an extraterrestrial presence as any president in history,” Holcombe writes.

In 1974, while governor of California, Reagan was flying on a Cessna Citation when he and others on board noticed a bright light that suddenly “shot up at a 45-degree angle at a high rate of speed.”

Years later, the pilot on the flight, Bill Paynter, “was asked if he thought Reagan believed in UFOs and he replied, ‘How could he not believe after what we saw that night?’”

Holcombe also cites a story told by actress Shirley MacLaine, who supposedly heard it from Lucille Ball. (Again, consider the source.)

En route to a party in LA, Reagan and his wife, Nancy, “encountered a UFO on the ground blocking their car. An alien exited the craft and addressed Reagan telepathically with one message. ‘Leave acting and go into politics.’ ”

Yes, this story is insane — but Holcombe argues that both Ball and the late Steve Allen said that “when the Reagans appeared at the party late, they appeared shaken and Reagan said they had seen a UFO.”

The detail about the telepathic alien, alas, was missing from Ball and Allen’s original version.

Holcombe writes that President Reagan had such a strong interest in UFOs that his aides “worked hard to keep a lid on the president’s interest.”

Holcombe also cites a “leaked document” from 1942 in which Gen. George Marshall, addressing President Franklin Roosevelt, makes reference to a mysterious sighting of several aircraft: They “are in fact not earthly, and according to secret intelligence sources they are in all probability of interplanetary origin.”

Amazing if true, but Holcombe says the memo’s authenticity is still debated.

John F. KennedyPhoto: AP

Just when you think this book can’t get any more bonkers, Holcombe writes that UFOs and aliens may have played a part in the Kennedy assassination.

Holcombe believes that Kennedy, who “requested UFO data from the CIA . . . had a great concern that UFOs could be mistaken by the US or the Soviet Union as a nuclear attack, and it is believed he discussed this issue with [Soviet leader Nikita] Khrushchev.”

Holcombe cites a classified CIA communication from the Kennedy era that reads, in part, “when conditions become nonconductive for growth in our environment and Washington cannot be influenced any further . . . it should be ‘wet.’ ”

Since “wet” was Soviet code for “assassination,” Holcombe thinks this could be CIA code for a plot to kill Kennedy. Why? He was considering telling the world that aliens existed, of course.

Not to worry: The Obama administration is on it. Appearing on “Jimmy Kimmel Live” last week, Obama was asked if he’d looked into Area 51, which has long been rumored to house the bodies of dead extraterrestrials.